## Method A: Clinometer

Measurements and Calculations:

**Procedure**

**1**. Choose a tree and stand at a distance

away from the tree where the top of the

tree can be seen

**2**. Direct Clinometer towards the top of the

Tree from the height of the eye

**3**. Record the angle observed while staying

stationary from the eye to the tip of tree

**4**. Record the distance at which you are

from the base of the tree

**5**. Record height of yourself from the base of your feet to your eyeball

**6**. Repeat steps 2-4 to maximize accuracy of results

**7**.Organize all data in a table

**8**. Create a diagram representing you using the clinometer to view the tip of the tree including all measurements recorded previously

**9**. Put your knowledge of trigonometric ratios to use to calculate the height of tree

**Results from Method 1**

The first method to measure the height of the tree utilized a basic clinometer. The same person stood at varying distances for three separate trials, resulting in the same height value but a difference in angles and distances from the tree. This led to the observation that as the angle of elevation increased (the angle between the horizontal line of sight and the top of the tree), the distance between the individual and the tree’s base decreased, indicating an inverse relationship between these two variables. **The average height of the tree from our values was found to be 550.4 cm or 5.5 metres.**

## Method B: Shadows

Measurements and Calculations:

**Procedure** **1.** Pick a tree and locate its shadow **2.** Measure the tree’s shadow from the base to the longest part of the shadow **3.** Ensure that you are on proper level footing for the next steps **4.** Measure your full height from toe to head **5.** After that, measure your shadow length from heel to longest point or your head on the shadow **6.** Repeat steps 3 to 5 for two more people with different heights **7.** Organize all the data in a table **8.** Create a diagram of two right-angled triangles (one for the tree, one for you) **9.** Use your knowledge of similar triangles to find the height of the tree

**Results from Method 2**

The second method utilized shadows and our knowledge of similar triangles to determine the height of the tree. Each member of the group stood at a uniform distance away from the tree, but because the height of each individual varied, this resulted in varying similar triangles that correspect with the tree.We observed that as the height of the individuals decreased, the length of their shadows also decreased, indicating a direct proportionality between these two variables . As the angle of sunlight remained constant, this allowed us to form similar triangles as the angles of the triangle remained constant.

- We calculated the average height of the

tree from method B to be 543.9 cm or

5.44 m.

**Method A: **Clinometer

Trial Number | Distance from Tree | Degree Angle from Eye | Height to Eye | Calculated height of tree |

1 | 6.15 metres | 32° | 167 cm | 551.3 cm |

2 | 4.10 metres | 44° | 167 cm | 562.9 cm |

3 | 1.57 metres | 67° | 167 cm | 536.9 cm |

**Method B: **Shadows

Trial Number | Height of Individual | Shadow Length Of Person | Shadow Length of Tree | Height of Tree |

1 | 172 cm | 92 cm | 290 cm | 541.7 cm |

2 | 178 cm | 96 cm | 290 cm | 537.6 cm |

3 | 170 cm | 89 cm | 290 cm | 522.4 cm |

**Comparing Results of Both Methods**:

After completing the purpose of this summative of calculating the height of trees by utilizing trigonometric ratios and similar triangles, method B seems to give more accurate values. With less variability within the results, utilizing the Shadow method provides a more reliable result than method A. As seen in the table for method A, on the third trial, with a recorded distance of much closer to the tree, it seems that the calculated height of the tree is much different and lower than the heights of the previous two trials.

When looking at the table for Method B, the results show a more reasonable spread at varying heights. This proves, in this case, results taken from similar triangles are more accurate/reliable. Though, to determine which method truly is better, specific circumstances and limitations need to be taken into account. Clinometers lean towards the side of providing accuracy, but the shadow method provides simplicity and accessibility, and given optimal lighting conditions, results can be just as, or if not better than those of clinometers.

Additionally, it’s worth noting that combining the two methods can help cross-validate the results and reduce potential errors. If feasible, measurements using both the shadow method and the clinometer method can provide a more comprehensive and reliable estimation of the tree’s height.